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Quantum Theory of the Chemisorptive Bond for Metals
VIII Quantum Theory of the Chemisorptive Bond for Metals
The hypothetical lattice model of the last chapter brings out many of the characteristics of chemisorption on solid surfaces. But there is, of course, a limit to its usefulness in explaining real systems. Each of the four types of solids—molecular crystals, covalent crystals, ionic crystals, and metallic conductors—has its peculiarities, and within each of these broad types, many differences exist in composition and kinds of imperfections. No general theory encompasses all situations. To progress, we must make allowances for the structural peculiarities of real solids. In this chapter, we shall be concerned with metals. The approach remains basic and only occasionally will reach that degree of detail necessary for the consideration of particular metals. We must be content for the present if such basic theories can make general distinctions in the nature of chemisorption on the various classes of solids. The description of electron levels in solids in terms of allowed and forbidden zones is applied to all periodic systems, metallic or otherwise. But a metal, or more specifically, a conductor, has a half-filled band, overlapping of completely full and completely empty bands, or more complex combinations, which account for electrical conductivity. Each atom of a metal is regarded as ionized in a sea of electrons. Unlike heteropolar crystals, where the individual units are more or less distinct positive and nega137
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8. QUANTUM THEORY OF THE CHEMISORPTIVE BOND FOR METALS
tive ions held together by electrostatic forces, the binding in metals is usually considered to be the result of an approximately uniform electron cloud. The characteristic electronic structure of metals profoundly influences the nature of chemisorption on their surfaces. The use of only one atomic orbital per metal atom, as in the previous chapter, precludes all effects of orbital overlapping so prevalent in metals. Transition metals bring up the additional question of the effect of empty d orbitals on chemisorption (Chapter IV). In Section 8.1, we summarize briefly those theories of metals which have been used in conjunction with studies of chemisorption, some of which have been discussed in earlier chapters. In Section 8.2, we discuss the theoretical studies of electronic states created by chemisorption on metal surfaces. As mentioned many times, the two basic approaches involve (1) the assumption of a localized bond and (2) treatment of the entire crystal plus foreign atom. In the latter approach, we have seen that both localized and nonlocalized states may be predicted. The localized approach has been treated in earlier chapters, but in this chapter we shall pursue further the concept of a localized surface compound, ionic and covalent, from a more fundamental point of view. In the last section, we shall discuss the electron density in a metal near a chemisorbed atom or molecule. Theoretical calculations suggest that electronic disturbances by a foreign atom near a metal surface are long-range, oscillatory, and nonisotropic. 8.1
Theories of Metals in Relation to Chemisorption
Every theory of chemisorption on metals envisions a picture of the substrate. These pictures range from oversimplified and qualitative to detailed and quantitative. One of the simplest theories of metals, the free-electron theory (Section 3.1), in which a metal is described as a potential box of. mobile valence electrons, leads to a theory of ionic chemisorption. The metal has no structure, its surface is smooth and nonperiodic, and adsorption is governed by the work function of the metal, the ionization potential or electron affinity of the adsorbate, and the mirrorimage potential between metal and adsorbate (Section 3.2). Another theory, the valence-bond theory, or rather a simplified empirical version of it (Section 2.2), assumes that d electrons in metals are localized either on atoms or in bonds and leads to qualitative theories of covalent chemisorption using d-orbital vacancies of transition metals (Chapter IV) or "dangling" sp and dsp hybridized orbitals emerging from the surface. In other studies, the theory of chemisorption itself defines a simplified
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THEORIES OF METALS IN RELATION TO CHEMISORPTION
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model of the metal. For example, when it is assumed that the foreign atom interacts with only one or a small cluster of surface metal atoms, the model of the metal is automatically limited to the number of metal atoms involved in the chemisorption process, although these few atoms are usually endowed with a metallic character. Chemisorption on small domains has been studied by various quantum-mechanical procedures, including the simple diatomic molecule approach of Higuchi et al. [1] (Section 3.3), the extended Hückel LCAO-MO method [2] (Section 2.3), the three-particle perturbation approach of Jansen [3] for d metals, and the surface compound concept of Grimley [4] (Section 8.2). Results of broad interest develop when the metal is considered to be a semi-infinite periodic structure with which adsorbate particles interact. A few methods in this category are empirical, giving only a nebulous picture of the metal. For example, the Lennard-Jones [5] potential has been applied to interactions of foreign atoms with a metal. It consists of attractive terms related to the dispersion forces and derived by London using quantum-mechanical methods. According to London's theory, electrons in atoms and molecules are in continuous motion even in their ground states, and thus possess rapidly fluctuating dipole moments. The fleeting dipole moment in one atom perturbs a neighboring one, inducing a moment in it; and the temporary moment in the first atom and the moment induced in the second lead to an attractive force between them. Dipole-dipole interactions lead to attractive potentials which are proportional to the inverse sixth power of the distance between the two atoms. Occasionally, dipole-quadrupole and quadrupole-quadrupole interactions are also included, which vary inversely as the eighth and tenth power of the distance, respectively. Attractive potentials are combined with a repulsive potential, which is taken to vary inversely in the range of the ninth to the fourteenth power of the distance of separation, usually the twelfth power. The total potential between two atoms, including dipole-dipole and dipole-quadrupole attractive potentials, is expressed by ud = -(Ci/r*) - (C2/r*) + (C3/r12).
(1)
Less frequently, an exponential repulsive potential proposed by Born and Mayer [5, 6] is employed; it has the form Be~ar. The interaction of an adsorbed atom with the metal lattice is determined by pairwise summation of the interactions of the foreign atom with each of the atoms of the metal. The summation may be carried out directly or by integration. In many cases, the nearest pairwise interactions are summed directly and the remainder by integration. When integration is performed , the lattice is tacitly assumed to be a semi-infinite continuum
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QUANTUM THEORY OF THE CHEMISORPTIVE BOND FOR METALS
Fig. 8.1. Lennard-Jones 6-12 integrated pair potential for an adsorbed molecule.
with no periodic structure. Using only the dipole-dipole attractive term — Ci/r6 and the repulsive term Cz/r12, integration gives, for the LennardJones potential, Us(z) = (3\/3/2)€s[(zo/z) 9 - (V*) 3 ]
(2)
where z is the vertical distance of an adsorbed atom from the surface, v>B(zo) = 0, (dua/dz) = 0 at z = z*} and uB(z*) = — eB, the potential energy minimum. The potential curve is plotted schematically in Fig. 8.1. The parameters of Eq. (2), e8 and z0, which are equivalent to the use of the constants Ci and Cz, may be determined by comparison with measured physical properties, such as virial coefficients and viscosity data. Values of the equilibrium distance z* of the adsorbed molecule from the surface are often assumed on the basis of some reasonable argument, and values of the parameter Ci may be determined by theoretical equations developed by London [5, 7], Slater and Kirkwood [5, 8], and Mueller [5, 9]. The dipole-dipole attractive potential — Ci/r* of Eq. (1) represents weak, or van der Waals, adsorption, and fits more closely the category of physical adsorption than chemisorption. Application to metal surfaces is complicated by the nonlocalized character of the electrons in metals to which the approximations of the Lennard-Jones potential are not strictly amen-
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THEORIES OF METALS IN RELATION TO CHEMISORPTION
141
able. The empirical nature of the method leads only to a nebulous picture of the metal. For this reason, we shall not discuss it further. Olander [10] has used a similar procedure to describe the interaction of hydrogen atoms with nickel and copper. He replaces the LennardJones potential with a Morse potential, which is more realistic for chemisorptive binding on metals, and is represented by the function *(r) = D { e x p [ - (2m/r.) (r - r . ) ] - 2 e x p [ - (m/re) (r - r.)]}
(3)
where
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[12], Fajans [13], and Bent [14]. The novel feature of Johnson's work [11] is the definitive description of the metallic state by an interstitialelectron model which incorporates all the quantum-mechanical aspects of chemical binding, including electron correlation and spin. His work is based on Berlin's [15] analysis of the Hellmann-Feynman theorem, which points out that at equilibrium the classical Coulomb forces exerted by quantum-mechanical electron distribution on the nuclei must balance precisely the forces of nuclear-nuclear repulsion. Extension of the model to metals allows the treatment of metals in a qualitative or quantitative manner on an electrostatic basis. We summarize the basic postulates of the interstitial-electron model applied to metals: 1. In a metal, the valence electrons belong to the entire structure as itinerant electrons. 2. Interstices of the close-packed metal structure are taken as the location of electrons. Electron density is centered on an interstice. The binding regions for metal electrons may be considered to be the tetrahedral and octahedral interstices. 3. A preferential occupancy of octahedral or tetrahedral interstices depends on which interstice provides most effective screening from adjacent electrons by positive ion cores. 4. Electrons in adjacent interstices with minimum screening have opposite spins. 5. Before the number of electrons reaches 6, the d electrons are localized on the metal ion core. ■6. Itinerant electrons in interstices act as ligands and determine the degeneracy of the d electrons localized on the ion core and, thus, their magnetic properties. 7. For metallic properties, vacant interstices must exist in the metal structure. 8. The shape of the octahedral or tetrahedral binding region can change with occupancy, since repulsion exerted by electrons in adjacent interstices defines the boundary of the binding region. Mutual polarization of itinerant electrons and ion cores may also modify electron density and "shape of. the binding region." These postulates imply that in an array of metal ion cores, the maximum attraction of these cores by electrons is obtained when some degree of localization of the electron density in the interstices of the structure exists. A consideration of the geometry of the various close-packed metal structures, as well as of the patterns of interstices within the struc-
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ELECTRONIC STATES IN CHEMISORPTION ON METALS
143
tures, reveals that there are very important differences in interstice distances and arrangements. The theory gives an instantaneous picture in a dynamic situation of the electron density in tetrahedral and octahedral interstices, including both occupied and vacant interstices. The average itinerant electron density is accurately represented by the instantaneous picture, and a time average picture results when the total number of electrons in an interstice type is fractionally distributed over the total number of interstices of that type. Other models, such as the orbital constructions of Trost [16] and the crystal-field band structures proposed by Goodenough [17], which we shall consider in Chapter X, emphasize the spatial arrangement of d electrons in transition metals. But the present model considers the itinerant electrons as ligands, not associated with a single metal ion core, and thus is related only indirectly to the models of Trost and Goodenough. Models of localized binding such as that of Pauling (Section 2.2) are much further removed from the band theory than the interstitial model. Application of the interstitial-electron model to chemisorption is speculative and qualitative at present, and we shall not pursue the model further. We note that the interstitial disposition of electrons could have significant implications for chemisorption, and the model is worth pursuing. 8.2
Electronic States in Chemisorption on Metals
No conclusions can be reached yet about the nature of a chemisorptive bond between a particular metal and foreign atom as to whether it is localized or nonlocalized. We note that the energy of the intermetallic bond in a crystal lies in the range 5-10 eV, the dissociation energy of a simple diatomic molecule is of the same order of magnitude, and the energy of the chemisorptive bond ranges from 2 to 10 eV. It is difficult to decide, therefore, whether to treat the surface interaction as a perturbation of the crystal by the approaching foreign atom, or to consider it a "surface compound" which is perturbed by the rest of the crystal. It seems probable a priori that examples of both types of systems should exist. First we shall discuss examples of those systems of chemisorption on metals which might reasonably be described as surface compounds. Similar, though less fundamental examples, were discussed in Section 2.3. After that, a discussion of treatments involving the entire metal plus foreign atom follows. THE CONCEPT OF A SURFACE COMPOUND
The concept that a definite surface compound is formed in chemisorption is used in the interpretation of infrared spectra of molecules
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adsorbed on metals; for example, in the analysis of the linear and bridged forms of CO on nickel, Ni OsC—Ni
and
0=C
\
Ni
It is a consequence of the idea of "dangling" surface orbitals, which leads naturally to the formation of a surface compound. Yet, we must reconcile this model with the band theory of metals, which in many aspects has proven extremely useful. The strong overlapping of bands in metals places severe restrictions on the existence of discrete levels of surface compounds lying outside all of the metal band systems. Although the precise concept of a surface compound on a metal surface may run afoul of the band theory, it proves useful as an approximation. If d orbitals on a surface metal atom are coupled to orbitals of a foreign atom more strongly than to d orbitals on neighboring metal atoms, it is reasonable to adopt the concept of a surface compound perturbed by the rest of the metal. First, the discrete orbital energies of the surface compound would be calculated, and then allowances would be made for the interaction of the compound with the itinerant electron states of the surrounding metal. In the event that the interaction is weak, the energy levels of the surface compound that fall in the energy bands of the metal are merely shifted and broadened slightly. Originally discrete, they now become virtual levels lying within the energy bands of the metal. In this sense, the concept of a surface compound may be retained. As long as the interaction is weak, the energy of the virtual levels may be calculated approximately by assuming a surface compound. If there is no interaction at all between the surface compound and the rest of the metal, then the levels are discrete. But as we mentioned earlier, the probability of such levels decreases markedly with increasing overlapping of the energy bands of the metal. That a surface metal atom interacting with a foreign atom will become completely "demetalized" and lose its collective properties as it forms a nonmetallic molecule seems theoretically improbable, though definitive experiments have not been designed yet to give an unequivocal answer. If interaction between the surface compound and the itinerant electron states of the rest of the metal is strong, the levels are so broadened that they bear no resemblance to discrete levels, and the concept of a surface compound is worthless. We see that our concept of a surface compound differs from the concept of an ordinary molecule in two ways. First, the metal atom of the surface compound is also a part of the bulk metal, and so itinerant electrons of
8.2
145
ELECTRONIC STATES IN CHEMISORPTION ON METALS
27T
0.2
VACUUM LEVEL
FERMI LEVEL
1
T"
4
0
d BAND
-0.2
s BAND
5
NICKEL
Fig. 8.2. The energy bands of nickel, and the 5
the metal as well as electrons of the surface compound contribute to the bonding. Second, as a consequence of the coupling to the itinerant electron states, the surface compound may contain a nonintegral number of electrons. Only in the limit, when the wave functions of the surface compound drop to zero on all of the metal atoms except the central atom (or atoms) initially defined as part of the compound, does the surface compound become the exact counterpart of the simple, isolated molecule. To give the concept of a surface compound a fair trial, a system with low heat of chemisorption should be selected. Grimley [4] has selected the system carbon monoxide chemisorbed on nickel, which turns out to be a good one. Other systems of low heat of adsorption are CO on iron, cobalt, rhodium, palladium, and platinum. Although an accurate theory of any of these systems is still remote, sufficient information exists for the nickel-carbon monoxide system to permit significant calculations. Accurate wave functions (Hartree-Fock type) and energies are available for carbon monoxide [18], and these may be combined with recent band structure calculations for nickel [19, 20]. We start by comparing the orbital energies of electrons in CO and nickel (Fig.8.2). The calculations of Matheiss give the positions of the s and d bands of nickel relative to the Fermi level, and the position of the Fermi level relative to the vacuum level is fixed
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QUANTUM THEORY OF THE CHEMISORPTIVE BOND FOR METALS
by the work function. For our purposes, the position of the Fermi level may be considered independent of the crystal face, and is taken as the zero of energy. The energy of the first vacant level in CO [21], the 2ir level, is +0.44 hartree (1 hartree = 27.21 eV). The highest filled level, 5
8.2
147
ELECTRONIC STATES IN CHEMISORPTION ON METALS
77"'
m7T27T-
FERMI L E V E L —
0.2
\
/ / 1 / 1
IX^ 1
/ / / / /
OTT
der
-0.2
mCT-
5cr-
H-0.4
Fig. 8.3. The orbital energies of the surface compound formed when CO is adsorbed on nickel. The unperturbed do- and dir orbital energ'es coincide with the Fermi evel. Energy scale in hartrees (4).
between zero and two when it lies exactly at the Fermi level. Consequently, we see that the energies and the occupancies of the orbitals of the surface compound must be determined together self-consistently. Electron interaction in the surface compound is the result of electron interaction in the CO molecule and the metal atom separately, as well as the result of CO-Ni coupling. These changes upon compound formation, according to Grimley, cause a small shift of electrons from the nickel atom to the CO molecule, so that the CO molecule now has 2.284 electrons, an increase of 0.284, and the nickel atom has 5.426 in do- and d7r orbitals, a decrease of 0.214. The final result of the quantum-mechanical calculations shows that the 5(T and 2ir orbitals of an isolated CO molecule are raised when the surface compound is formed, and the do- and dir orbitals of the nickel atom are lowered. In the diagram in Fig. 8.3, these shifted levels are shown as mc and mir for CO and do- and dx for Ni, the latter having shifted downward from the Fermi level. The diagram indicates how these shifted levels interact to form the levels (a- and a*) and (w and T*) of the surface compound. So far, the calculations do not include the perturbation of the surface compound by the itinerant electron states of the metal. As a first approximation, Grimley assumes that only the sp band of nickel is involved in the coupling of the surface compound, for this band is much broader than the d band, whose energies lie within it. Once this coupling is triggered, all of the calculated discrete levels of the surface compound become virtual levels, and the number of electrons associated with the unperturbed surface compound are changed slightly. From quantum-mechanical theory,
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QUANTUM THEORY OF THE CHEMISORPTIVE BOND FOR METALS
Grimley concludes that such changes can only raise the total energy ( < 0 ) of the compound, which means that the binding of CO to nickel can only decrease as a result of including perturbations of the surface compound by the itinerant electron states of the metal. Since the binding energy of the unperturbed surface compound Ni-CO is calculated to be 0.0981 hartree, the binding energy of the perturbed compound will be slightly lower than that value. Another localized approach has been proposed by Jansen [3]—a threeparticle perturbation calculation for chemisorption on d-electron metals. Binding is assumed to occur through latent d-electron free valency of the metal ions, and the role of the conduction (itinerant electrons) of the metal is nonspecific and, like Grimley's approach, is treated as a perturbation. Jansen gives several reasons why chemisorption of a molecule on a single metal atom is not reasonable, among them the fact that chemisorption varies with crystallographic plane—a phenomenon which makes little sense if chemisorption is restricted to a single metal atom. The next simplest approximation is the consideration of the three-particle system, two metal ions and a chemisorbed particle. Moreover, Jansen selects the simplest three-particle system: two metal ions, each with one unpaired electron, and an adsorbate atom or molecule with two paired electrons. The theory predicts strong chemisorption of hydrogen molecules on a platinum surface, but cannot explain their dissociation on the surface. If a four-particle system is considered, the theory does predict dissociation [24]. The theory is not as well developed as Grimley's theory, and its value is difficult to assess at its present stage. Accounts of other theories of localized adsorption may be found in Sections 2.3 and 3.3 related to covalent and ionic bonding, respectively, and in Section 7.6 concerning the use of small finite models. To these we should add a more detailed and specific calculation for hydrogen adsorption on a nickel crystal by Fassaert et al. [25]. These authors use the extended Hückel MO method and treat adsorption on Ni (111), (100), and (110). With a cluster of 13 nickel atoms having metallic properties, they found that the adsorption energy decreases with increasing number of neighbors, which bears a similarity to Grimley's [ 4 ] finding of highest adsorption energy when perturbations of the surface compound by itinerant electrons of the metal lattice are ignored. TREATMENTS INVOLVING THE ENTIRE CRYSTAL PLUS FOREIGN ATOM
Ultimately, methods in this category should be the most rewarding, for in them is the latent power to decide for a specific system the nature
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ELECTRONIC STATES IN CHEMISORPTION ON METALS
149
of the chemisorption bond from the one extreme of discrete bonding to the other of metallic bonding. Yet enthusiasm must be tempered with an appreciation of the complexities. The simpler localized approaches of the preceding subsection will remain useful for some time to come. We start with a discussion of calculations by Grimley, who treated the adsorption of an atom on nickel by the familiar LCAO self-consistent band picture, and extracted the concept of a surface molecule as the limiting case of the general theory. The treatment differs from his earlier work [4], described above, on the Ni-CO system, which starts with the assumption of a surface compound and modifies it by a small perturbation. Which of the two approaches to the concept of a surface molecule is to be preferred will depend on the metal-adsorbate system under consideration. One of the specific systems considered by Grimley [26] is the chemisorption of an atom on the (100) face of nickel. He uses the value of 0.349 Ry (1 rydberg = 13.54 eV) from the work of Gerlach and Rhodin [27] for the work function of the clean surface, and this fixes the Fermi level, which is set at energy zero. Calculations on the free Na atom set the energy of the 3s valence orbital at —0.015 Ry below the Fermi level. At first thought, we would not expect the Na atom to lose its valence electron to the metal. But as we have already seen, the situation changes as the atom approaches the metal. The discrete atomic level is transformed into a virtual level with some broadening and with part of its density above the Fermi level. Thus the 3s orbital has a different degree of occupancy in the adsorbed atom (no occupancy above the Fermi level). A self-consistent calculation for the degree of occupancy and the orbital energy of the adsorbed atom must be made, and in this case Grimley uses the general approach, considering all the metal atoms plus the sodium atom. For this purpose, he employs a model Hamiltonian of the type first used by Anderson [28] in the theory of dilute alloys. It comprises four terms, two of which describe the free atom, one the metal, and the fourth the coupling between them. There are obvious objections
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QUANTUM THEORY OF THE CHEMISORPTIVE BOND FOR METALS
face compound is found to differ from an ordinary molecule, for it is an open system and need not contain an integral number of electrons. Orbital energy and occupancy of the orbital of adsorbed Na were calculated as a function of the coupling parameter. In order to estimate a reasonable value of the coupling parameter, the experimental value of the heat of adsorption (0.2 Ry) was used. On this basis, the orbital energy of electrons in the surface bond was calculated to be —0.237 Ry. Gadzuk et al [29] have performed similar calculations for the chemisorption of alkali metal atoms on metals, using the Anderson model Hamiltonian. In considering the nature of the chemisorptive bond, they find the major portion of the binding is ionic. Values of binding energies in the range 1.5-2.5 eV were calculated, which, they cite, agree with experimental values for heats of chemisorption on tungsten, molybdenum, tantalum, and nickel. We note that Grimley [26] quoted an experimental value of 0.2 Ry ( = 2.7 eV), which is in reasonable agreement with the range of experimental and theoretical values given by Gadzuk. A value of the orbital energy of electrons in the surface bond is not available for comparison with Grimley's value of 0.237 Ry ( = - 3 . 2 5 eV). Grimley [30] has also studied the chemisorption of the hydrogen atom on metals. He considers first anionic adsorption, in which an electron is transferred from the metal to the atom, giving a negative hydrogen ion at the surface. The interaction energy Q ( = — AH) is given by an equation of the form Q = -4> + A0 + K + J
(4)
where 4> is the work function of the metal, A0 the electron affinity of the hydrogen atom, and K and J the Coulomb and the quantum-mechanical exchange (resonance) interaction, respectively, of H~ with the metal. A simplified form of Eq. (4) neglects J and replaces K by the familiar mirrorimage potential (see Section 3.2). The sum ( — <£ + A0) is negative for all metals, which means that anionic interaction is endothermic, unless counteracted by {K + «/), and therefore not likely to occur. Grimley [30] sets up wave functions and derives energy expressions for K and «/, representing interaction of H~ with the metal. He concludes that both terms involve only short-range interactions. Further, he presents evidence that the short-range interaction energies of such systems are negative. Thus the complete expression for Q in Eq. (4) is always negative and anionic chemisorption of hydrogen atoms is not favored. A similar conclusion was reached in Section 3.2, using the simplified form of Eq. (4) with the mirror-image potential in place of (K + J). Next, Grimley considers the formation of a covalent bond. He uses an
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ELECTRONIC STATES IN CHEMISORPTION ON METALS
151
LCAO procedure similar to the one described in Sections 7.1 and 7.2, and comes to the same conclusion, namely, that provided surface states exist for the isolated metal, localized states exist for the combined system. If there is a localized state with energy lying below the Fermi level in the metal, the corresponding orbital will be occupied by two electrons of opposite spin, and something like a covalent bond will be formed between hydrogen and metal. As mentioned earlier, such a bond will be an open system, and in all probability will have a nonintegral number of electrons. Since a localized level must lie outside of all metal band systems, it is a difficult problem to determine just where, below the Fermi level, this localized state might be situated. A localized, covalent bond lying below the metal band structure should represent the important binding of a hydrogen atom to a metal surface. However, the existence of virtual levels at a higher energy level and within the band system, as described in the Ni-CO system, cannot be definitely excluded, though the binding energy would be less. In a review article by Horiuti and Toya [31], two types of chemisorption of hydrogen atoms on metals, r and s, are distinguished, and LCAOMO calculations made for each type. The r adatom is a chemisorbed hydrogen atom of conventional character, located outside the electron cloud of the metal surface and directly above a surface metal atom. The s atom is interstitially located between the electronic surface and the layer of surface metal atoms, nearly embedded in the electron cloud. They are illustrated in Fig. 8.4. The r adatom in its equilibrium position is slightly negatively polarized, and thus this type of adsorption increases the work function of the metal. The authors claim that electrical resistance measurements on nickel films with adsorbed hydrogen are in conflict with the concept of the formation of a localized bond, and they propose that the bond is metallic in nature. The s adatom may be thought of as an atom dissolved in the metal and dissociated into a proton and an electron in the con4uçtkm band. The equilibrium position and the corresponding energy of the PFQton have been determined, using equations developed by Thomas [32], Fermi [33], and Weizsäcker [34]. The energy in tfee equilibrium position was found to be lower than that of the dissölyeä state in the interior of the metal by 0.5 eV for nickel and 1.0 e y for platinum. This adatom is associated with a small positive dipole mqrnènt and is not bound to a particular metal atom. It may be viewed as the special case of a proton dissolved in the surface, and differs from adsorption models considered previously. In concluding this section, we merely cite a paper by May and Carrol [35] that deals with ionic monolayers on metals and discusses the recon-
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8. QUANTUM THEORY OF THE CHEMISORPTIVE BOND FOR METALS
ELECTRONIC SURFACE-^
r-TYPE ADSORPTION
o o„îo O
s-TYPE ADSORPTION
ION
( a ) SIDE VIEW
© o o r-TYPE ADSORPTION
O s - T Y P E ADSORPTION
O OuO ' METAL ION
( b ) TOP VIEW
Fig. 8.4- The equilibrium of an r-type adatom above a metal ion, ^ 1 A outside the electronic surface of the metal, and that of an s-type adatom interstitial, M).5 A inside the electronic surface, (a) Side view; (b) top view (31). [After Horiuti and Tomiyuki, "Solid State Surface Science" (Mino Green, ed.), Vol. 1, Marcel Dekker, New York 1969. By permission of Marcel Dekker Publishing Company.]
struction of surfaces, and one by Grimley [36] that describes the chemisorption of a hydrogen atom on tungsten (100) as the surface complex W2H. 8.3
Electron Densities
It was demonstrated some time ago [37, 38] that an impurity atom in a metal causes localized perturbations in electron density, long-range and oscillatory in nature. Grimley [39] speculated that a chemisorbed atom should produce similar perturbations, since it is an impurity atom on the surface. There are no experimental investigations of this phenomenon reported, and indeed, it is not yet clear exactly how to design a significant experiment. Grimley's speculations and calculations are important, for they point to interesting possibilities, which require corroboration or denial, and offer a challenge to experimentalists. A simple LCAO-MO model is used, similar to the general model of Chapter VII, except that Wannier functions are employed instead of the more usual atomic wave functions. The wave function of an electron in a
8.3
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ELECTRON DENSITIES
crystal may be expressed as a linear combination of Wannier functions which are approximately localized in the regions of the individual elementary cells (single atom per cell). A formal similarity exists between Wannier functions and the atomic orbitals of the LCAO-MO tight binding approximation. The electron density at a point is designated as p(r) and is obtained by summing the square of the complete wave function, | ^(r) |2, for the crystal plus chemisorbed atom. If r is near a lattice point, we expect the main contribution to p(r) from the wave function of the atom at that point. This approximation justifies the use of Wannier functions. Without the chemisorbed atom, the occupancy of metal orbitals should be the same at all lattice points in the metal surface. The presence of the chemisorbed atom disturbs this situation. Grimley has made calculations with and without chemisorption for several cases: 1. The system, crystal plus chemisorbed atom, has a virtual level near the Fermi level. (See Section 8.2 for the description of a virtual level associated with chemisorption.) 2. The system has a virtual level near the bottom of the conduction band. 3. The system has a discrete level. In general, the calculations show that the disturbance is anisotropic. When Ä, the distance from a surface crystal atom, is in the critical direction, the disturbance falls away more rapidly than in other directions. For a body-centered cubic metal with a (110) surface, the critical direction is the (110) direction. Each of the three foregoing cases has a different expression for the electron density as a function of Ä, both for the critical direction and other directions, though approximately all vary as Ä"5 and Ä~3, respectively. For the chemisorption of alkali metal atoms on tungsten or nickel, a virtual level near the Fermi level probably exists, and the critical direction is (110). On this basis, Grimley has calculated the change in occupancy, An(Ä), of an orbital level on a surface metal atom at a distance R from the chemisorbed atom. The function An(R) bears an obvious relationship to p(r). In noncritical directions, the change in occupancy is given by An(R) = 0.036(sin 2.8Ä/Ä3),
R in A;
(5)
and in the critical direction, An(R) = -0.059(sin2.8Ä/ß 5 ).
(6)
Thus the simple model predicts a disturbance near a chemisorbed atom that is anisotropic, long range, and oscillatory.
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8.
QUANTUM THEORY OF THE CHEMISORPTIVE BOND FOR METALS
The model has certain weaknesses, as do all simplifications of the complex real picture. In particular, Coulomb interactions need further study. Only Coulomb interactions between electrons in the chemisorbed atom were included, while those terms with both electrons in the metal, or one in the metal and one on the atom, were neglected. Problems associated with Coulomb interactions and effective shielding remain intractable in the theory of chemisorption. Until these problems are solved, no theory of chemisorption may be considered entirely satisfactory. REFERENCES 1. I. Higuchi, T. Ree, and H. Eyring, / . Amer. Chem. Soc. 79, 1330 (1957). 2. J. C. Robertson and C. W. Wilmsen, J. Vac. Sei. Technol. 8, 53 (1971). 3. L. Jansen, Mol. Proc. Solid Surfaces, Battelle Inst. Mater. Sei. Colloq., 3rd p. 49 (1968). 4. T. B. Grimley, Mol. Proc. Solid Surfaces, Battelle Inst. Mater. Sei. Colloq., 3rd p. 299 (1968). 5. A. Clark, "The Theory of Adsorption and Catalysis," p. 141. Academic Press, New York, 1970. 6. M. Born and J. E. Mayer, Z. Phys. 75,1 (1932). 7. J. C. London, Z. Phys. 63, 245 (1930). 8. J. C. Slater and J. G. Kirkwood, Phys. Rev. 37, 682 (1931). 9. A. Mueller, Proc. Roy. Soc. London A154, 624 (1936). 10. D. R. Olander, J. Phys. Chem. Solids 32, 2499 (1971). 11. O. Johnson, Bull. Chem. Soc. Japan 45, 1599, 1607 (1972); J. Res. Inst. Catal. Hokkaido 19, 152(1972). 12. H. A. Lorentz, "Theory of Electrons." Teubner, Leipzig, 1906. 13. K. Fajans, Ceram. Age 54, 288 (1949). 14. H. Bent, / . Chem. Educ. 42, 348 (1965). 15. T. Berlin, / . Chem. Phys 19,208(1951). 16. W. A. Trost, Can. J. Chem. 37, 466 (1959). 17. J. B. Goodenough, Phys. Rev. 120, 67 (1966). 18. W. M. Huo, J. Chem. Phys. 43, 624 (1965). 19. L. F. Matheiss, Phys. Rev. A134, 970 (1964). 20. S. Wakok and J. Yamashita, / . Phys. Soc. Japan 19,1342 (1964). 21. R. C. Sahni, C. D. la Budde, and B. C. Sawney, Trans. Faraday Soc. 62,1993 (1966). 22. D. Brennan and F. H. Hages, Phil. Trans. A258, 347 (1965). 23. L. E. Orgel, "An Introduction to Transition-Metal Chemistry, Ligand-Field Theory." Wiley, New York and Methuen, London (1960). 24. A. van der Avoird, Thesis, Tech. Univ. Eindhoven, December, 1968. 25. D. J. M. Fassaert, H. Verbeek, and A. van der Avoird, Surface Sei. 29, 501 (1972). 26. T. B. Grimley, J. Vac. Sei. Technol. 8, 31 (1971). 27. R. Gerlach and T. N. Rhodin, "The Structure and Chemistry of Solid Surfaces" (G. A. Somorjai, ed.), p. 55. Wiley, New York, 1969. 28. P. W. Anderson, Phys. Rev. 124, 41 (1961).
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155
29. J. W. Gadzuk, J. K. Hartmann, and T. N. Rhodin, Phys. Rev. B 4, 241 (1971). 30. T. B. Grimley, in "Chemisorption" (W. E. Garner, ed.). Butterworth, London and Washington D.C., 1957. 31. J. Horiuti and Toya Tomiyuki, "Solid State Surface Science" (M. Green, ed.), Vol. 1. Dekker, New York, 1969. 32. L. H. Thomas, Proc. Cambridge Phil Soc. 23, 542 (1927). 33. E. Fermi, Z. Phys. 48, 73 (1928). 34. C. F. von Weizsäcker, Z. Phys. 96, 431 (1935). 35. J. W. May and C. E. Carroll, Surface Sei. 29, 60, 85 (1972). 36. T. B. Grimley, Ber. Bunsenges. Phys. Chem. 75, 1003 (1971). 37. J. Friedel, Advan. Phys. 3,446 (1954). 38. Yu. A. Izyumov, Advan. Phys. 14, 569 (1965). 39. T. B. Grimley, Proc. Phys. Soc. London 92, 776 (1967).
The hypothetical lattice model of the last chapter brings out many of the characteristics of chemisorption on solid surfaces. But there is, of course, a limit to its usefulness in explaining real systems. Each of the four types of solids—molecular crystals, covalent crystals, ionic crystals, and metallic conductors—has its peculiarities, and within each of these broad types, many differences exist in composition and kinds of imperfections. No general theory encompasses all situations. To progress, we must make allowances for the structural peculiarities of real solids. In this chapter, we shall be concerned with metals. The approach remains basic and only occasionally will reach that degree of detail necessary for the consideration of particular metals. We must be content for the present if such basic theories can make general distinctions in the nature of chemisorption on the various classes of solids. The description of electron levels in solids in terms of allowed and forbidden zones is applied to all periodic systems, metallic or otherwise. But a metal, or more specifically, a conductor, has a half-filled band, overlapping of completely full and completely empty bands, or more complex combinations, which account for electrical conductivity. Each atom of a metal is regarded as ionized in a sea of electrons. Unlike heteropolar crystals, where the individual units are more or less distinct positive and nega137
138
8. QUANTUM THEORY OF THE CHEMISORPTIVE BOND FOR METALS
tive ions held together by electrostatic forces, the binding in metals is usually considered to be the result of an approximately uniform electron cloud. The characteristic electronic structure of metals profoundly influences the nature of chemisorption on their surfaces. The use of only one atomic orbital per metal atom, as in the previous chapter, precludes all effects of orbital overlapping so prevalent in metals. Transition metals bring up the additional question of the effect of empty d orbitals on chemisorption (Chapter IV). In Section 8.1, we summarize briefly those theories of metals which have been used in conjunction with studies of chemisorption, some of which have been discussed in earlier chapters. In Section 8.2, we discuss the theoretical studies of electronic states created by chemisorption on metal surfaces. As mentioned many times, the two basic approaches involve (1) the assumption of a localized bond and (2) treatment of the entire crystal plus foreign atom. In the latter approach, we have seen that both localized and nonlocalized states may be predicted. The localized approach has been treated in earlier chapters, but in this chapter we shall pursue further the concept of a localized surface compound, ionic and covalent, from a more fundamental point of view. In the last section, we shall discuss the electron density in a metal near a chemisorbed atom or molecule. Theoretical calculations suggest that electronic disturbances by a foreign atom near a metal surface are long-range, oscillatory, and nonisotropic. 8.1
Theories of Metals in Relation to Chemisorption
Every theory of chemisorption on metals envisions a picture of the substrate. These pictures range from oversimplified and qualitative to detailed and quantitative. One of the simplest theories of metals, the free-electron theory (Section 3.1), in which a metal is described as a potential box of. mobile valence electrons, leads to a theory of ionic chemisorption. The metal has no structure, its surface is smooth and nonperiodic, and adsorption is governed by the work function of the metal, the ionization potential or electron affinity of the adsorbate, and the mirrorimage potential between metal and adsorbate (Section 3.2). Another theory, the valence-bond theory, or rather a simplified empirical version of it (Section 2.2), assumes that d electrons in metals are localized either on atoms or in bonds and leads to qualitative theories of covalent chemisorption using d-orbital vacancies of transition metals (Chapter IV) or "dangling" sp and dsp hybridized orbitals emerging from the surface. In other studies, the theory of chemisorption itself defines a simplified
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THEORIES OF METALS IN RELATION TO CHEMISORPTION
139
model of the metal. For example, when it is assumed that the foreign atom interacts with only one or a small cluster of surface metal atoms, the model of the metal is automatically limited to the number of metal atoms involved in the chemisorption process, although these few atoms are usually endowed with a metallic character. Chemisorption on small domains has been studied by various quantum-mechanical procedures, including the simple diatomic molecule approach of Higuchi et al. [1] (Section 3.3), the extended Hückel LCAO-MO method [2] (Section 2.3), the three-particle perturbation approach of Jansen [3] for d metals, and the surface compound concept of Grimley [4] (Section 8.2). Results of broad interest develop when the metal is considered to be a semi-infinite periodic structure with which adsorbate particles interact. A few methods in this category are empirical, giving only a nebulous picture of the metal. For example, the Lennard-Jones [5] potential has been applied to interactions of foreign atoms with a metal. It consists of attractive terms related to the dispersion forces and derived by London using quantum-mechanical methods. According to London's theory, electrons in atoms and molecules are in continuous motion even in their ground states, and thus possess rapidly fluctuating dipole moments. The fleeting dipole moment in one atom perturbs a neighboring one, inducing a moment in it; and the temporary moment in the first atom and the moment induced in the second lead to an attractive force between them. Dipole-dipole interactions lead to attractive potentials which are proportional to the inverse sixth power of the distance between the two atoms. Occasionally, dipole-quadrupole and quadrupole-quadrupole interactions are also included, which vary inversely as the eighth and tenth power of the distance, respectively. Attractive potentials are combined with a repulsive potential, which is taken to vary inversely in the range of the ninth to the fourteenth power of the distance of separation, usually the twelfth power. The total potential between two atoms, including dipole-dipole and dipole-quadrupole attractive potentials, is expressed by ud = -(Ci/r*) - (C2/r*) + (C3/r12).
(1)
Less frequently, an exponential repulsive potential proposed by Born and Mayer [5, 6] is employed; it has the form Be~ar. The interaction of an adsorbed atom with the metal lattice is determined by pairwise summation of the interactions of the foreign atom with each of the atoms of the metal. The summation may be carried out directly or by integration. In many cases, the nearest pairwise interactions are summed directly and the remainder by integration. When integration is performed , the lattice is tacitly assumed to be a semi-infinite continuum
140
8.
QUANTUM THEORY OF THE CHEMISORPTIVE BOND FOR METALS
Fig. 8.1. Lennard-Jones 6-12 integrated pair potential for an adsorbed molecule.
with no periodic structure. Using only the dipole-dipole attractive term — Ci/r6 and the repulsive term Cz/r12, integration gives, for the LennardJones potential, Us(z) = (3\/3/2)€s[(zo/z) 9 - (V*) 3 ]
(2)
where z is the vertical distance of an adsorbed atom from the surface, v>B(zo) = 0, (dua/dz) = 0 at z = z*} and uB(z*) = — eB, the potential energy minimum. The potential curve is plotted schematically in Fig. 8.1. The parameters of Eq. (2), e8 and z0, which are equivalent to the use of the constants Ci and Cz, may be determined by comparison with measured physical properties, such as virial coefficients and viscosity data. Values of the equilibrium distance z* of the adsorbed molecule from the surface are often assumed on the basis of some reasonable argument, and values of the parameter Ci may be determined by theoretical equations developed by London [5, 7], Slater and Kirkwood [5, 8], and Mueller [5, 9]. The dipole-dipole attractive potential — Ci/r* of Eq. (1) represents weak, or van der Waals, adsorption, and fits more closely the category of physical adsorption than chemisorption. Application to metal surfaces is complicated by the nonlocalized character of the electrons in metals to which the approximations of the Lennard-Jones potential are not strictly amen-
8.1
THEORIES OF METALS IN RELATION TO CHEMISORPTION
141
able. The empirical nature of the method leads only to a nebulous picture of the metal. For this reason, we shall not discuss it further. Olander [10] has used a similar procedure to describe the interaction of hydrogen atoms with nickel and copper. He replaces the LennardJones potential with a Morse potential, which is more realistic for chemisorptive binding on metals, and is represented by the function *(r) = D { e x p [ - (2m/r.) (r - r . ) ] - 2 e x p [ - (m/re) (r - r.)]}
(3)
where
142
8.
QUANTUM THEORY OF THE CHEMISORPTIVE BOND FOR METALS
[12], Fajans [13], and Bent [14]. The novel feature of Johnson's work [11] is the definitive description of the metallic state by an interstitialelectron model which incorporates all the quantum-mechanical aspects of chemical binding, including electron correlation and spin. His work is based on Berlin's [15] analysis of the Hellmann-Feynman theorem, which points out that at equilibrium the classical Coulomb forces exerted by quantum-mechanical electron distribution on the nuclei must balance precisely the forces of nuclear-nuclear repulsion. Extension of the model to metals allows the treatment of metals in a qualitative or quantitative manner on an electrostatic basis. We summarize the basic postulates of the interstitial-electron model applied to metals: 1. In a metal, the valence electrons belong to the entire structure as itinerant electrons. 2. Interstices of the close-packed metal structure are taken as the location of electrons. Electron density is centered on an interstice. The binding regions for metal electrons may be considered to be the tetrahedral and octahedral interstices. 3. A preferential occupancy of octahedral or tetrahedral interstices depends on which interstice provides most effective screening from adjacent electrons by positive ion cores. 4. Electrons in adjacent interstices with minimum screening have opposite spins. 5. Before the number of electrons reaches 6, the d electrons are localized on the metal ion core. ■6. Itinerant electrons in interstices act as ligands and determine the degeneracy of the d electrons localized on the ion core and, thus, their magnetic properties. 7. For metallic properties, vacant interstices must exist in the metal structure. 8. The shape of the octahedral or tetrahedral binding region can change with occupancy, since repulsion exerted by electrons in adjacent interstices defines the boundary of the binding region. Mutual polarization of itinerant electrons and ion cores may also modify electron density and "shape of. the binding region." These postulates imply that in an array of metal ion cores, the maximum attraction of these cores by electrons is obtained when some degree of localization of the electron density in the interstices of the structure exists. A consideration of the geometry of the various close-packed metal structures, as well as of the patterns of interstices within the struc-
8.2
ELECTRONIC STATES IN CHEMISORPTION ON METALS
143
tures, reveals that there are very important differences in interstice distances and arrangements. The theory gives an instantaneous picture in a dynamic situation of the electron density in tetrahedral and octahedral interstices, including both occupied and vacant interstices. The average itinerant electron density is accurately represented by the instantaneous picture, and a time average picture results when the total number of electrons in an interstice type is fractionally distributed over the total number of interstices of that type. Other models, such as the orbital constructions of Trost [16] and the crystal-field band structures proposed by Goodenough [17], which we shall consider in Chapter X, emphasize the spatial arrangement of d electrons in transition metals. But the present model considers the itinerant electrons as ligands, not associated with a single metal ion core, and thus is related only indirectly to the models of Trost and Goodenough. Models of localized binding such as that of Pauling (Section 2.2) are much further removed from the band theory than the interstitial model. Application of the interstitial-electron model to chemisorption is speculative and qualitative at present, and we shall not pursue the model further. We note that the interstitial disposition of electrons could have significant implications for chemisorption, and the model is worth pursuing. 8.2
Electronic States in Chemisorption on Metals
No conclusions can be reached yet about the nature of a chemisorptive bond between a particular metal and foreign atom as to whether it is localized or nonlocalized. We note that the energy of the intermetallic bond in a crystal lies in the range 5-10 eV, the dissociation energy of a simple diatomic molecule is of the same order of magnitude, and the energy of the chemisorptive bond ranges from 2 to 10 eV. It is difficult to decide, therefore, whether to treat the surface interaction as a perturbation of the crystal by the approaching foreign atom, or to consider it a "surface compound" which is perturbed by the rest of the crystal. It seems probable a priori that examples of both types of systems should exist. First we shall discuss examples of those systems of chemisorption on metals which might reasonably be described as surface compounds. Similar, though less fundamental examples, were discussed in Section 2.3. After that, a discussion of treatments involving the entire metal plus foreign atom follows. THE CONCEPT OF A SURFACE COMPOUND
The concept that a definite surface compound is formed in chemisorption is used in the interpretation of infrared spectra of molecules
144
8.
QUANTUM THEORY OF THE CHEMISORPTIVE BOND FOR METALS
adsorbed on metals; for example, in the analysis of the linear and bridged forms of CO on nickel, Ni OsC—Ni
and
0=C
\
Ni
It is a consequence of the idea of "dangling" surface orbitals, which leads naturally to the formation of a surface compound. Yet, we must reconcile this model with the band theory of metals, which in many aspects has proven extremely useful. The strong overlapping of bands in metals places severe restrictions on the existence of discrete levels of surface compounds lying outside all of the metal band systems. Although the precise concept of a surface compound on a metal surface may run afoul of the band theory, it proves useful as an approximation. If d orbitals on a surface metal atom are coupled to orbitals of a foreign atom more strongly than to d orbitals on neighboring metal atoms, it is reasonable to adopt the concept of a surface compound perturbed by the rest of the metal. First, the discrete orbital energies of the surface compound would be calculated, and then allowances would be made for the interaction of the compound with the itinerant electron states of the surrounding metal. In the event that the interaction is weak, the energy levels of the surface compound that fall in the energy bands of the metal are merely shifted and broadened slightly. Originally discrete, they now become virtual levels lying within the energy bands of the metal. In this sense, the concept of a surface compound may be retained. As long as the interaction is weak, the energy of the virtual levels may be calculated approximately by assuming a surface compound. If there is no interaction at all between the surface compound and the rest of the metal, then the levels are discrete. But as we mentioned earlier, the probability of such levels decreases markedly with increasing overlapping of the energy bands of the metal. That a surface metal atom interacting with a foreign atom will become completely "demetalized" and lose its collective properties as it forms a nonmetallic molecule seems theoretically improbable, though definitive experiments have not been designed yet to give an unequivocal answer. If interaction between the surface compound and the itinerant electron states of the rest of the metal is strong, the levels are so broadened that they bear no resemblance to discrete levels, and the concept of a surface compound is worthless. We see that our concept of a surface compound differs from the concept of an ordinary molecule in two ways. First, the metal atom of the surface compound is also a part of the bulk metal, and so itinerant electrons of
8.2
145
ELECTRONIC STATES IN CHEMISORPTION ON METALS
27T
0.2
VACUUM LEVEL
FERMI LEVEL
1
T"
4
0
d BAND
-0.2
s BAND
5
NICKEL
Fig. 8.2. The energy bands of nickel, and the 5
the metal as well as electrons of the surface compound contribute to the bonding. Second, as a consequence of the coupling to the itinerant electron states, the surface compound may contain a nonintegral number of electrons. Only in the limit, when the wave functions of the surface compound drop to zero on all of the metal atoms except the central atom (or atoms) initially defined as part of the compound, does the surface compound become the exact counterpart of the simple, isolated molecule. To give the concept of a surface compound a fair trial, a system with low heat of chemisorption should be selected. Grimley [4] has selected the system carbon monoxide chemisorbed on nickel, which turns out to be a good one. Other systems of low heat of adsorption are CO on iron, cobalt, rhodium, palladium, and platinum. Although an accurate theory of any of these systems is still remote, sufficient information exists for the nickel-carbon monoxide system to permit significant calculations. Accurate wave functions (Hartree-Fock type) and energies are available for carbon monoxide [18], and these may be combined with recent band structure calculations for nickel [19, 20]. We start by comparing the orbital energies of electrons in CO and nickel (Fig.8.2). The calculations of Matheiss give the positions of the s and d bands of nickel relative to the Fermi level, and the position of the Fermi level relative to the vacuum level is fixed
146
8.
QUANTUM THEORY OF THE CHEMISORPTIVE BOND FOR METALS
by the work function. For our purposes, the position of the Fermi level may be considered independent of the crystal face, and is taken as the zero of energy. The energy of the first vacant level in CO [21], the 2ir level, is +0.44 hartree (1 hartree = 27.21 eV). The highest filled level, 5
8.2
147
ELECTRONIC STATES IN CHEMISORPTION ON METALS
77"'
m7T27T-
FERMI L E V E L —
0.2
\
/ / 1 / 1
IX^ 1
/ / / / /
OTT
der
-0.2
mCT-
5cr-
H-0.4
Fig. 8.3. The orbital energies of the surface compound formed when CO is adsorbed on nickel. The unperturbed do- and dir orbital energ'es coincide with the Fermi evel. Energy scale in hartrees (4).
between zero and two when it lies exactly at the Fermi level. Consequently, we see that the energies and the occupancies of the orbitals of the surface compound must be determined together self-consistently. Electron interaction in the surface compound is the result of electron interaction in the CO molecule and the metal atom separately, as well as the result of CO-Ni coupling. These changes upon compound formation, according to Grimley, cause a small shift of electrons from the nickel atom to the CO molecule, so that the CO molecule now has 2.284 electrons, an increase of 0.284, and the nickel atom has 5.426 in do- and d7r orbitals, a decrease of 0.214. The final result of the quantum-mechanical calculations shows that the 5(T and 2ir orbitals of an isolated CO molecule are raised when the surface compound is formed, and the do- and dir orbitals of the nickel atom are lowered. In the diagram in Fig. 8.3, these shifted levels are shown as mc and mir for CO and do- and dx for Ni, the latter having shifted downward from the Fermi level. The diagram indicates how these shifted levels interact to form the levels (a- and a*) and (w and T*) of the surface compound. So far, the calculations do not include the perturbation of the surface compound by the itinerant electron states of the metal. As a first approximation, Grimley assumes that only the sp band of nickel is involved in the coupling of the surface compound, for this band is much broader than the d band, whose energies lie within it. Once this coupling is triggered, all of the calculated discrete levels of the surface compound become virtual levels, and the number of electrons associated with the unperturbed surface compound are changed slightly. From quantum-mechanical theory,
148
8.
QUANTUM THEORY OF THE CHEMISORPTIVE BOND FOR METALS
Grimley concludes that such changes can only raise the total energy ( < 0 ) of the compound, which means that the binding of CO to nickel can only decrease as a result of including perturbations of the surface compound by the itinerant electron states of the metal. Since the binding energy of the unperturbed surface compound Ni-CO is calculated to be 0.0981 hartree, the binding energy of the perturbed compound will be slightly lower than that value. Another localized approach has been proposed by Jansen [3]—a threeparticle perturbation calculation for chemisorption on d-electron metals. Binding is assumed to occur through latent d-electron free valency of the metal ions, and the role of the conduction (itinerant electrons) of the metal is nonspecific and, like Grimley's approach, is treated as a perturbation. Jansen gives several reasons why chemisorption of a molecule on a single metal atom is not reasonable, among them the fact that chemisorption varies with crystallographic plane—a phenomenon which makes little sense if chemisorption is restricted to a single metal atom. The next simplest approximation is the consideration of the three-particle system, two metal ions and a chemisorbed particle. Moreover, Jansen selects the simplest three-particle system: two metal ions, each with one unpaired electron, and an adsorbate atom or molecule with two paired electrons. The theory predicts strong chemisorption of hydrogen molecules on a platinum surface, but cannot explain their dissociation on the surface. If a four-particle system is considered, the theory does predict dissociation [24]. The theory is not as well developed as Grimley's theory, and its value is difficult to assess at its present stage. Accounts of other theories of localized adsorption may be found in Sections 2.3 and 3.3 related to covalent and ionic bonding, respectively, and in Section 7.6 concerning the use of small finite models. To these we should add a more detailed and specific calculation for hydrogen adsorption on a nickel crystal by Fassaert et al. [25]. These authors use the extended Hückel MO method and treat adsorption on Ni (111), (100), and (110). With a cluster of 13 nickel atoms having metallic properties, they found that the adsorption energy decreases with increasing number of neighbors, which bears a similarity to Grimley's [ 4 ] finding of highest adsorption energy when perturbations of the surface compound by itinerant electrons of the metal lattice are ignored. TREATMENTS INVOLVING THE ENTIRE CRYSTAL PLUS FOREIGN ATOM
Ultimately, methods in this category should be the most rewarding, for in them is the latent power to decide for a specific system the nature
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ELECTRONIC STATES IN CHEMISORPTION ON METALS
149
of the chemisorption bond from the one extreme of discrete bonding to the other of metallic bonding. Yet enthusiasm must be tempered with an appreciation of the complexities. The simpler localized approaches of the preceding subsection will remain useful for some time to come. We start with a discussion of calculations by Grimley, who treated the adsorption of an atom on nickel by the familiar LCAO self-consistent band picture, and extracted the concept of a surface molecule as the limiting case of the general theory. The treatment differs from his earlier work [4], described above, on the Ni-CO system, which starts with the assumption of a surface compound and modifies it by a small perturbation. Which of the two approaches to the concept of a surface molecule is to be preferred will depend on the metal-adsorbate system under consideration. One of the specific systems considered by Grimley [26] is the chemisorption of an atom on the (100) face of nickel. He uses the value of 0.349 Ry (1 rydberg = 13.54 eV) from the work of Gerlach and Rhodin [27] for the work function of the clean surface, and this fixes the Fermi level, which is set at energy zero. Calculations on the free Na atom set the energy of the 3s valence orbital at —0.015 Ry below the Fermi level. At first thought, we would not expect the Na atom to lose its valence electron to the metal. But as we have already seen, the situation changes as the atom approaches the metal. The discrete atomic level is transformed into a virtual level with some broadening and with part of its density above the Fermi level. Thus the 3s orbital has a different degree of occupancy in the adsorbed atom (no occupancy above the Fermi level). A self-consistent calculation for the degree of occupancy and the orbital energy of the adsorbed atom must be made, and in this case Grimley uses the general approach, considering all the metal atoms plus the sodium atom. For this purpose, he employs a model Hamiltonian of the type first used by Anderson [28] in the theory of dilute alloys. It comprises four terms, two of which describe the free atom, one the metal, and the fourth the coupling between them. There are obvious objections
150
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QUANTUM THEORY OF THE CHEMISORPTIVE BOND FOR METALS
face compound is found to differ from an ordinary molecule, for it is an open system and need not contain an integral number of electrons. Orbital energy and occupancy of the orbital of adsorbed Na were calculated as a function of the coupling parameter. In order to estimate a reasonable value of the coupling parameter, the experimental value of the heat of adsorption (0.2 Ry) was used. On this basis, the orbital energy of electrons in the surface bond was calculated to be —0.237 Ry. Gadzuk et al [29] have performed similar calculations for the chemisorption of alkali metal atoms on metals, using the Anderson model Hamiltonian. In considering the nature of the chemisorptive bond, they find the major portion of the binding is ionic. Values of binding energies in the range 1.5-2.5 eV were calculated, which, they cite, agree with experimental values for heats of chemisorption on tungsten, molybdenum, tantalum, and nickel. We note that Grimley [26] quoted an experimental value of 0.2 Ry ( = 2.7 eV), which is in reasonable agreement with the range of experimental and theoretical values given by Gadzuk. A value of the orbital energy of electrons in the surface bond is not available for comparison with Grimley's value of 0.237 Ry ( = - 3 . 2 5 eV). Grimley [30] has also studied the chemisorption of the hydrogen atom on metals. He considers first anionic adsorption, in which an electron is transferred from the metal to the atom, giving a negative hydrogen ion at the surface. The interaction energy Q ( = — AH) is given by an equation of the form Q = -4> + A0 + K + J
(4)
where 4> is the work function of the metal, A0 the electron affinity of the hydrogen atom, and K and J the Coulomb and the quantum-mechanical exchange (resonance) interaction, respectively, of H~ with the metal. A simplified form of Eq. (4) neglects J and replaces K by the familiar mirrorimage potential (see Section 3.2). The sum ( — <£ + A0) is negative for all metals, which means that anionic interaction is endothermic, unless counteracted by {K + «/), and therefore not likely to occur. Grimley [30] sets up wave functions and derives energy expressions for K and «/, representing interaction of H~ with the metal. He concludes that both terms involve only short-range interactions. Further, he presents evidence that the short-range interaction energies of such systems are negative. Thus the complete expression for Q in Eq. (4) is always negative and anionic chemisorption of hydrogen atoms is not favored. A similar conclusion was reached in Section 3.2, using the simplified form of Eq. (4) with the mirror-image potential in place of (K + J). Next, Grimley considers the formation of a covalent bond. He uses an
8.2
ELECTRONIC STATES IN CHEMISORPTION ON METALS
151
LCAO procedure similar to the one described in Sections 7.1 and 7.2, and comes to the same conclusion, namely, that provided surface states exist for the isolated metal, localized states exist for the combined system. If there is a localized state with energy lying below the Fermi level in the metal, the corresponding orbital will be occupied by two electrons of opposite spin, and something like a covalent bond will be formed between hydrogen and metal. As mentioned earlier, such a bond will be an open system, and in all probability will have a nonintegral number of electrons. Since a localized level must lie outside of all metal band systems, it is a difficult problem to determine just where, below the Fermi level, this localized state might be situated. A localized, covalent bond lying below the metal band structure should represent the important binding of a hydrogen atom to a metal surface. However, the existence of virtual levels at a higher energy level and within the band system, as described in the Ni-CO system, cannot be definitely excluded, though the binding energy would be less. In a review article by Horiuti and Toya [31], two types of chemisorption of hydrogen atoms on metals, r and s, are distinguished, and LCAOMO calculations made for each type. The r adatom is a chemisorbed hydrogen atom of conventional character, located outside the electron cloud of the metal surface and directly above a surface metal atom. The s atom is interstitially located between the electronic surface and the layer of surface metal atoms, nearly embedded in the electron cloud. They are illustrated in Fig. 8.4. The r adatom in its equilibrium position is slightly negatively polarized, and thus this type of adsorption increases the work function of the metal. The authors claim that electrical resistance measurements on nickel films with adsorbed hydrogen are in conflict with the concept of the formation of a localized bond, and they propose that the bond is metallic in nature. The s adatom may be thought of as an atom dissolved in the metal and dissociated into a proton and an electron in the con4uçtkm band. The equilibrium position and the corresponding energy of the PFQton have been determined, using equations developed by Thomas [32], Fermi [33], and Weizsäcker [34]. The energy in tfee equilibrium position was found to be lower than that of the dissölyeä state in the interior of the metal by 0.5 eV for nickel and 1.0 e y for platinum. This adatom is associated with a small positive dipole mqrnènt and is not bound to a particular metal atom. It may be viewed as the special case of a proton dissolved in the surface, and differs from adsorption models considered previously. In concluding this section, we merely cite a paper by May and Carrol [35] that deals with ionic monolayers on metals and discusses the recon-
152
8. QUANTUM THEORY OF THE CHEMISORPTIVE BOND FOR METALS
ELECTRONIC SURFACE-^
r-TYPE ADSORPTION
o o„îo O
s-TYPE ADSORPTION
ION
( a ) SIDE VIEW
© o o r-TYPE ADSORPTION
O s - T Y P E ADSORPTION
O OuO ' METAL ION
( b ) TOP VIEW
Fig. 8.4- The equilibrium of an r-type adatom above a metal ion, ^ 1 A outside the electronic surface of the metal, and that of an s-type adatom interstitial, M).5 A inside the electronic surface, (a) Side view; (b) top view (31). [After Horiuti and Tomiyuki, "Solid State Surface Science" (Mino Green, ed.), Vol. 1, Marcel Dekker, New York 1969. By permission of Marcel Dekker Publishing Company.]
struction of surfaces, and one by Grimley [36] that describes the chemisorption of a hydrogen atom on tungsten (100) as the surface complex W2H. 8.3
Electron Densities
It was demonstrated some time ago [37, 38] that an impurity atom in a metal causes localized perturbations in electron density, long-range and oscillatory in nature. Grimley [39] speculated that a chemisorbed atom should produce similar perturbations, since it is an impurity atom on the surface. There are no experimental investigations of this phenomenon reported, and indeed, it is not yet clear exactly how to design a significant experiment. Grimley's speculations and calculations are important, for they point to interesting possibilities, which require corroboration or denial, and offer a challenge to experimentalists. A simple LCAO-MO model is used, similar to the general model of Chapter VII, except that Wannier functions are employed instead of the more usual atomic wave functions. The wave function of an electron in a
8.3
153
ELECTRON DENSITIES
crystal may be expressed as a linear combination of Wannier functions which are approximately localized in the regions of the individual elementary cells (single atom per cell). A formal similarity exists between Wannier functions and the atomic orbitals of the LCAO-MO tight binding approximation. The electron density at a point is designated as p(r) and is obtained by summing the square of the complete wave function, | ^(r) |2, for the crystal plus chemisorbed atom. If r is near a lattice point, we expect the main contribution to p(r) from the wave function of the atom at that point. This approximation justifies the use of Wannier functions. Without the chemisorbed atom, the occupancy of metal orbitals should be the same at all lattice points in the metal surface. The presence of the chemisorbed atom disturbs this situation. Grimley has made calculations with and without chemisorption for several cases: 1. The system, crystal plus chemisorbed atom, has a virtual level near the Fermi level. (See Section 8.2 for the description of a virtual level associated with chemisorption.) 2. The system has a virtual level near the bottom of the conduction band. 3. The system has a discrete level. In general, the calculations show that the disturbance is anisotropic. When Ä, the distance from a surface crystal atom, is in the critical direction, the disturbance falls away more rapidly than in other directions. For a body-centered cubic metal with a (110) surface, the critical direction is the (110) direction. Each of the three foregoing cases has a different expression for the electron density as a function of Ä, both for the critical direction and other directions, though approximately all vary as Ä"5 and Ä~3, respectively. For the chemisorption of alkali metal atoms on tungsten or nickel, a virtual level near the Fermi level probably exists, and the critical direction is (110). On this basis, Grimley has calculated the change in occupancy, An(Ä), of an orbital level on a surface metal atom at a distance R from the chemisorbed atom. The function An(R) bears an obvious relationship to p(r). In noncritical directions, the change in occupancy is given by An(R) = 0.036(sin 2.8Ä/Ä3),
R in A;
(5)
and in the critical direction, An(R) = -0.059(sin2.8Ä/ß 5 ).
(6)
Thus the simple model predicts a disturbance near a chemisorbed atom that is anisotropic, long range, and oscillatory.
154
8.
QUANTUM THEORY OF THE CHEMISORPTIVE BOND FOR METALS
The model has certain weaknesses, as do all simplifications of the complex real picture. In particular, Coulomb interactions need further study. Only Coulomb interactions between electrons in the chemisorbed atom were included, while those terms with both electrons in the metal, or one in the metal and one on the atom, were neglected. Problems associated with Coulomb interactions and effective shielding remain intractable in the theory of chemisorption. Until these problems are solved, no theory of chemisorption may be considered entirely satisfactory. REFERENCES 1. I. Higuchi, T. Ree, and H. Eyring, / . Amer. Chem. Soc. 79, 1330 (1957). 2. J. C. Robertson and C. W. Wilmsen, J. Vac. Sei. Technol. 8, 53 (1971). 3. L. Jansen, Mol. Proc. Solid Surfaces, Battelle Inst. Mater. Sei. Colloq., 3rd p. 49 (1968). 4. T. B. Grimley, Mol. Proc. Solid Surfaces, Battelle Inst. Mater. Sei. Colloq., 3rd p. 299 (1968). 5. A. Clark, "The Theory of Adsorption and Catalysis," p. 141. Academic Press, New York, 1970. 6. M. Born and J. E. Mayer, Z. Phys. 75,1 (1932). 7. J. C. London, Z. Phys. 63, 245 (1930). 8. J. C. Slater and J. G. Kirkwood, Phys. Rev. 37, 682 (1931). 9. A. Mueller, Proc. Roy. Soc. London A154, 624 (1936). 10. D. R. Olander, J. Phys. Chem. Solids 32, 2499 (1971). 11. O. Johnson, Bull. Chem. Soc. Japan 45, 1599, 1607 (1972); J. Res. Inst. Catal. Hokkaido 19, 152(1972). 12. H. A. Lorentz, "Theory of Electrons." Teubner, Leipzig, 1906. 13. K. Fajans, Ceram. Age 54, 288 (1949). 14. H. Bent, / . Chem. Educ. 42, 348 (1965). 15. T. Berlin, / . Chem. Phys 19,208(1951). 16. W. A. Trost, Can. J. Chem. 37, 466 (1959). 17. J. B. Goodenough, Phys. Rev. 120, 67 (1966). 18. W. M. Huo, J. Chem. Phys. 43, 624 (1965). 19. L. F. Matheiss, Phys. Rev. A134, 970 (1964). 20. S. Wakok and J. Yamashita, / . Phys. Soc. Japan 19,1342 (1964). 21. R. C. Sahni, C. D. la Budde, and B. C. Sawney, Trans. Faraday Soc. 62,1993 (1966). 22. D. Brennan and F. H. Hages, Phil. Trans. A258, 347 (1965). 23. L. E. Orgel, "An Introduction to Transition-Metal Chemistry, Ligand-Field Theory." Wiley, New York and Methuen, London (1960). 24. A. van der Avoird, Thesis, Tech. Univ. Eindhoven, December, 1968. 25. D. J. M. Fassaert, H. Verbeek, and A. van der Avoird, Surface Sei. 29, 501 (1972). 26. T. B. Grimley, J. Vac. Sei. Technol. 8, 31 (1971). 27. R. Gerlach and T. N. Rhodin, "The Structure and Chemistry of Solid Surfaces" (G. A. Somorjai, ed.), p. 55. Wiley, New York, 1969. 28. P. W. Anderson, Phys. Rev. 124, 41 (1961).
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ELECTRON DENSITIES
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29. J. W. Gadzuk, J. K. Hartmann, and T. N. Rhodin, Phys. Rev. B 4, 241 (1971). 30. T. B. Grimley, in "Chemisorption" (W. E. Garner, ed.). Butterworth, London and Washington D.C., 1957. 31. J. Horiuti and Toya Tomiyuki, "Solid State Surface Science" (M. Green, ed.), Vol. 1. Dekker, New York, 1969. 32. L. H. Thomas, Proc. Cambridge Phil Soc. 23, 542 (1927). 33. E. Fermi, Z. Phys. 48, 73 (1928). 34. C. F. von Weizsäcker, Z. Phys. 96, 431 (1935). 35. J. W. May and C. E. Carroll, Surface Sei. 29, 60, 85 (1972). 36. T. B. Grimley, Ber. Bunsenges. Phys. Chem. 75, 1003 (1971). 37. J. Friedel, Advan. Phys. 3,446 (1954). 38. Yu. A. Izyumov, Advan. Phys. 14, 569 (1965). 39. T. B. Grimley, Proc. Phys. Soc. London 92, 776 (1967).